Question

If $\overline{a},\overline{b},\overline{c}$ are three vectors such that $\overline{a}+\overline{b}+\overline{c}=\overline{0}$, then

• A. $\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{c}\times \overline{a}$
• B. $\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{c}\times \overline{a}=0$
• C. $\overline{a}\times \overline{b}=\overline{b}\times \overline{c}$
• D. $\overline{a}\times \overline{b}=0$

Answer 1

The correct option is A $\overline{a}\times \overline{b}=\overline{b}\times \overline{c}=\overline{c}\times \overline{a}$

1. $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$

vector multiply by $\overrightarrow{a}$
$\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}=0$
$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{a}$ .........(1)
vector multiply by $\overrightarrow{b}$
$\overrightarrow{b}\times \overrightarrow{a}+\overrightarrow{b}\times \overrightarrow{c}=0$
$\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}$........(2)
Equate (1) & (2)
$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{c}$